First of all, I like to use a container with a tight-fitting lid. A short, wide jar, such as some brands of salsa come in, will work perfectly, but just about any jar will do. Between coats you can close the jar and put the paint in the refrigerator.
Milk paint, once mixed with water, has a short shelf life, maybe 48 hours tops, so it makes sense to mix small batches, ideally just enough for your current project. Begin by measuring some water into the jar -- for a Windsor chair I find that about 1/2 cup is a good start. Add paint powder about 1/2 tablespoon at a time. Mix until all the powder disappears into the water, then add some more. Don't worry about lumps at this point. Continue adding paint and stirring it in until the mixture begins to thicken.
If there aren't any lumps in the mixture (not likely) simply stop when the paint is at the desired thickness. To get rid of lumps, continue to add powder until the paint becomes overly thick. As the stirring stick pushes those lumps through the thick paint they will be squeezed away -- that's the theory anyway, and it seems to work. If the lumps are persistence, make it even thicker. Finally, if some just won't go away, forget them; they will settle to the bottom and your brush won't find them until you reach the bottom of the batch.
Now that you have paint that is too thick to use, begin adding water, a little at a time, until you reach the desired thickness. What is desired? Probably a little thinner than other paints you've used. But the real test is to try it out. If it's difficult to spread, add more water.
After the first coat, twist the jar-lid on snuggly. If it will be a while before you apply the next coat (a few hours, overnight...), put the jar in the refrigerator.
I use a good quality, polyester bristle brush. Dip the dry brush into water before starting, then shake out the excess. Wetting the brush helps prevent paint drying in the upper part of the bristles. (This is a good idea for any kind of painting. For oil-based paints dip into mineral spirits.)
The first coat won't flow on as easily as you might expect. When it dries, it will be somewhat transparent and full of overlaps. At this point you may feel you have embarked on a misadventure, but do not despair.
Before the second coat, you may want to rub down the surface with steel-wool or a Scotchbrite scouring pad (available at the hardware or paint store, or your grocery store). I usually just seek out the roughtest spots and deal with those for now, sometimes even lightly sanding (very lightly) with fine sandpaper.
The second coat will flow on very nicely. The paint will spread farther and dry faster. With the darker colors -- black, the reds, the greens -- lap-marks will no longer be an issue, unless you are working on a large surface, such as a table top, where it is difficult to maintain a "wet edge." Also, some colors are more transparent than others. The blues are somewhat transparent and it pays to watch out for lapping while you apply the second coat. Yellow and orange are especially so and may even require a third coat. Even with three coats the yellows (mustard, marigold) may prove transparent when oiled.
After all the coats are applied, give the whole thing a good rubbing with steel wool (#000 or #0000) or a scouring pad, then vacuum off the dust. You are now ready for oil.
I use boiled linseed oil cut with a little turpentine, about 6 to 1. The turpentine is used as a drying agent. Spread the oil mixture on liberally with a foam brush. When everything is coated, go over those areas that have dried. Let it sit for 15 or 20 minutes then touch up the dry areas once more. Give it another half-hour or so then wipe away all the excess oil. I use cheese cloth for this, also available at your grocery store and maybe the hardware store. They may leave a little lint behind, but after the piece has dried for a couple hours, this can be vacuumed way.
The result will be a matte finish. If you want a little more "shine" let it dry a few days then re-oil. For a real gloss use an oil/varnish mixture such as "Antique Oil," from Minwax or Watco Danish Oil. But don't leave it on too long or you may get more gloss than you intended. Experimenting on scrap wood is recommended.
Your project may have a piney, turpentine smell for a few days, but it will be pretty much dry in about 24 hours.
A milk paint/linseed finish is susceptible to water spots. If a spill is wiped up right away there likely will be no problem. But if left to dry, whiteish (cloudy) spots will result. These are easily removed with a little oil/turp, rubbed in and wiped away.
The only thing left is to enjoy the result. Good luck!
I know safety warnings have become so common as to sometimes seem meaningless, but I am begging you to heed this one. Please, please, please dispose of all oily rags or paper towels properly. A byproduct of the drying process is heat. On an open surface this heat quickly dissipates into the air is never detected. But it will accumulate in a balled-up rag in a waste basket or trash barrel to the point of spontaneous combustion. More than one house/shop/garage has burned down from carelessly disposed-of oily rags.
The simplest measure is to take the rags outside, away from the house, and lay them out flat or drape them over something -- a tree branch, a wheel barrow handle, a trash barrel lid. Once completely dry, they can be tossed in the trash (but still consider keeping them away from the house).
If putting the oily rags outside is not practical, place them in a bucket of water.
For more information about milk paint, and to find a retailer in your area, see the Old Fashioned Milk Paint Company.